Elastic Compression of Spheres and Cylinders
at Point and Line Contact
Jack A. Stone and Jay H. Zimmerman
Abstract: These web pages are intended primarily as a computational tool that can be used to calculate the elastic compression of spheres and cylinders at point and line contact.1
Whenever two surfaces are in contact under force, the surfaces elastically deform and compress. This web page calculates the amount of compression at point and line contacts between spheres, cylinders, and planes, using formula developed by Puttock and Thwaite1. This deformation must be taken into account in all high-accuracy measurements when a surface is probed mechanically. The web page is intended to serve as an aid to people involved in high-accuracy measurements where deformation corrections are essential in order to achieve reliable measurement results.
Please note the following important warnings when using this web page.
- Elastic Limit: The program assumes elastic deformation of the contact and does not test to see if the elastic limit of the materials is exceeded. Consequently, at high force on small contact area, the results will not give the true deformation of the material. The user of this web page must carry out independent checks to determine if the elastic limit is exceeded.
- Material Properties: The elastic properties of materials used in the deformation calculations are only approximate values. For example, the Young's Modulus given for Aluminum is characteristic of common alloys, but different alloys may differ in their elastic properties by 10% or more. For any material shown, it may be expected that uncertainties in the elastic properties are of this magnitude. If better accuracy is needed and you know the elastic coefficients of your particular alloy, select Other [Please Specify] for the material and enter your known elastic coefficients.
- Other Material Properties: If you choose a specific material from the list of materials, but input elastic coefficients, these inputs will be ignored; they are only used when you select Other [Please Specify].
- Additional Limitations: Puttock and Thwaite also warn that their formula may not be valid at low force, particularly when surface finish is bad. (Their formula assume that the surfaces are perfectly smooth.) They also assume that frictional forces at the contact point are negligible, and that the materials in contact are homogeneous.
- Rounding Error: Results are expressed in both metric (SI) and English units. These results may disagree at the sub-nanometer level (0.01 microinch) as a consequence of rounding.
II. Using the Web Page
Select the Geometry Cases web page, which has hyperlinks to the sixteen geometric arrangements or cases. (Note that we have not yet implemented all sixteen cases.) Then select the desired contact geometry, enter the correct information into the form (i.e., angle, applied force, diameter, length, material properties), and press Calculate Elastic Compression. The web page will re-display the information that was entered into the form along with the calculated deformation. Results of the calculation are followed by a new form to re-enter information for another calculation.
Note that the form provides accepted entry values, expressed in accepted (SI) units, that must be changed as needed by the user of the web page. For known material properties, select Other [Please Specify], re-enter values for Young's Modulus or Poisson's Ratio that the web page will use for a calculation. Otherwise, the web page reassigns accepted values to Young's Modulus and Poisson's Ratio for materials selected.
III. Background Information
The sixteen geometric arrangements or cases use formulae developed and published in 1969 by Puttock and Thwaite for the Division of Applied Physics in the National Standards Laboratory at the Commonwealth Scientific and Industrial Research Organization (CSIRO) in Australia. These formula solve for Hertzian compression effects assuming
- Applied forces are within the elasticity limits of the materials
- Contact surfaces are perfectly smooth (finely lapped)
- Materials are of homogeneous composition
- Frictional forces at contact point are negligible.
1. M.J. Puttock and E.G. Thwaite, "Elastic Compression of Spheres and Cylinders at Point and Line Contact," National Standards Laboratory Technical Paper No. 25, Division of Appled Phyics, National Standards Laboratory, Commonwealth Scientific and Industrial Research Organization (CSIRO), University Grounds, Chippendale, New South Wales, Australia 2008, 1969.
Key words: elastic compression
Place of Publication: electronic, NIST web page